Transcendental Kähler Cohomology Classes

Abstract

Associated with a real, smooth, $d$-closed $(1,1)$-form $\alpha$ of possibly non-rational de Rham cohomology class on a compact complex manifold $X$ is a sequence of asymptotically holomorphic complex line bundles $L_k$ on $X$ equipped with $(0,1)$-connections ${\bar{\partial }}_k$ for which ${\bar{\partial }}_k^2\ne 0$. Their study was begun in the thesis of L. Laeng. We propose in this non-integrable context a substitute for Hörmander's familiar $L^2$-estimates of the $\bar{\partial }$-equation of the integrable case that is based on analysing the spectra of the Laplace–Beltrami operators ${\Delta }_k^{\prime \prime }$ associated with ${\bar{\partial }}_k$. Global approximately holomorphic peak sections of $L_k$ are constructed as a counterpart to Tian's holomorphic peak sections of the integral-class case. Two applications are then obtained when $\alpha$ is strictly positive: a Kodaira-type approximately holomorphic projective embedding theorem and a Tian-type almost-isometry theorem for compact Kähler, possibly non-projective, manifolds. Unlike in similar results in the literature for symplectic forms of integral classes, the peculiarity of $\alpha$ lies in its transcendental class. This approach will be hopefully continued in future work by relaxing the positivity assumption on $\alpha$.